How to pass a soft object through a narrow channel?
What you've posted there is no soft object passing through a channel, it's actually a Cylinder with a Lattice Modifier moving through a Lattice object.

Add a Cylinder to your scene or whatever object you want to deform.
To be deformed smoothly it needs enough geometry, so subdivide it either in Edit Mode or with a Subdivision Surface modifier.
Add a Lattice to your scene (Shift+A &gt; Lattice)
In the Lattice settings, set the Resolution W to at least 3.
Go to Edit Mode and scale the middle vertices down so that they narrow the Lattice in the center.
Add a Lattice Modifier to the Cylinder and under Object choose the Lattice.
Now you can move the Cylinder up and down and it gets deformed.

EDIT: I'm too slow. The above solution is correct, this is just a shorter version.